Exponential Integral Calculator

Published on: July 17, 2026
Final Answer: Free • Full Steps: Plus

This Exponential Integral Calculator integrates functions where the variable sits in the exponent, and shows the working line by line. The natural exponential integrates to itself, and any other base picks up a division by the natural logarithm of that base. That division is the mirror image of the factor that appears when differentiating. Following the same pattern every time keeps the base factor from being dropped.

Step-by-step method

  1. Set up the problem.
  2. Write the integration rule for that exponential function.
  3. Apply the rule, keeping the constant multiple in front, and add C.

Formula:

\(\begin{gathered} \int e^{x}\, dx = e^{x} + C \\ \int a^{x}\, dx = \frac{a^{x}}{\ln\left(a\right)} + C \end{gathered}\)
See Example 1 Hide Example 1

Problem

\(\int e^{x}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the integration rule for that exponential function.
  3. Apply the rule, keeping the constant multiple in front, and add C.

Step 1

Step 1 - Identify the exponential function and any constant multiple.

In this problem: The function is \(e^{x}\).

\(\int e^{x}\, dx\)

Step 2

Step 2 - Write the integration rule for that exponential function.

In this problem: The integral of \(e^{x}\) is itself.

\(\int e^{x}\, dx = e^{x} + C\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front, and add C.

In this problem: The antiderivative is \(e^{x}\).

\(\int e^{x}\, dx = e^{x} + C\)

Final Answer

\(\int e^{x}\, dx = e^{x} + C\)
See Example 2 Hide Example 2

Problem

\(\int 3^{x}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the integration rule for that exponential function.
  3. Apply the rule, keeping the constant multiple in front, and add C.

Step 1

Step 1 - Identify the exponential function and any constant multiple.

In this problem: The function is \(3^{x}\).

\(\int 3^{x}\, dx\)

Step 2

Step 2 - Write the integration rule for that exponential function.

In this problem: The integral of \(3^{x}\) is the function divided by the natural logarithm of the base \(3\).

\(\int a^{x}\, dx = \frac{a^{x}}{\ln\left(a\right)} + C\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front, and add C.

In this problem: The antiderivative is \(\frac{3^{x}}{\ln\left(3\right)}\).

\(\int 3^{x}\, dx = \frac{3^{x}}{\ln\left(3\right)} + C\)

Final Answer

\(\int 3^{x}\, dx = \frac{3^{x}}{\ln\left(3\right)} + C\)

Example Problems

Click one to load it into the box and solve it.